While true, I do believe that there is a merit to having at least one or two bike divisions.The idea has potential but the issue I have with it is that the niche it has is occupied by cavalry and the fact that we may have to build a supply line to build a supply line for the bicycles, which we may not have the resources to do.
While yes, the same niche is occupied by cavalry and cavalry would be faster, bikes overall would be cheaper and work better in terrain that horses couldn't be used or it would be difficult to use them in. The biggest example is the Japanese invasion of Malaya/Singapore, they were able to quickly and quietly advance in such a way that it caught the British defenders off guard with seized bikes from the local population.
I would also like to point out that maintaining a bicycle is much easier and costs far less than keeping horses.
Overall, bikes shouldn't be all that expensive when you compare them to guns, artillery or trucks and while the first few hundred or so would be purely for the army, we could then sell to the general public which could help out with the budget or at the very least recoup our investment.